Learning quantum states of continuous variable systems (2405.01431v4)

Abstract: Quantum state tomography, aimed at deriving a classical description of an unknown state from measurement data, is a fundamental task in quantum physics. In this work, we analyse the ultimate achievable performance of tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of finite-dimensional systems: not only does the minimum number of state copies needed for tomography scale exponentially with the number of modes, but it also exhibits a dramatic scaling with the trace-distance error, even for low-energy states, in stark contrast with the finite-dimensional case. On a more positive note, we prove that tomography of Gaussian states is efficient. To accomplish this, we answer a fundamental question for the field of continuous-variable quantum information: if we know with a certain error the first and second moments of an unknown Gaussian state, what is the resulting trace-distance error that we make on the state? Lastly, we demonstrate that tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.

• The paper analyzes the sample complexity for quantum state tomography in continuous-variable systems, establishing bounds for pure and mixed states under energy constraints.
• It demonstrates that efficient tomography is feasible for Gaussian states by measuring their first moments and covariance matrices via homodyne detection.
• The study extends to non-Gaussian states, showing efficient tomography for t-doped Gaussian states prepared with limited non-Gaussian operations.

Overview of "Learning Quantum States of Continuous-Variable Systems"

The paper "Learning Quantum States of Continuous-Variable Systems" explores the complexities and methodologies associated with quantum state tomography in continuous-variable (CV) systems. It presents a comprehensive analysis of the techniques used to derive classical descriptions of quantum systems from experimental data. With a focus on continuous-variable systems, the research addresses a critical task in quantum information theory—quantifying the resources needed for accurate tomography.

The authors investigate the sample complexity required for quantum state tomography in both pure and mixed CV states under various energy constraints. They establish that for energy-constrained pure states, the sample complexity scales as $O(E^n/\varepsilon^{2n})$ where $E$ is the energy constraint, $n$ is the number of modes, and $\varepsilon$ is the trace-distance error. For mixed states, they find both upper and lower bounds on the sample complexity, ranging from $O(E^{2n}/\varepsilon^{3n})$ to $O(E^{2n}/\varepsilon^{2n})$. These results highlight the efficiency challenges and demonstrate that CV quantum state tomography is inherently more resource-intensive than its finite-dimensional counterpart.

An important contribution of this work is the focus on Gaussian states, which are central to many applications in quantum optics and continuous-variable quantum computing. The authors demonstrate that the efficient tomography of Gaussian states is feasible. They achieve this by establishing a bound on the trace distance between Gaussian states based on their first moments and covariance matrices. The technique relies on measuring these parameters with a high degree of accuracy using homodyne detection, thus allowing for efficient reconstruction of Gaussian states with a polynomial resource overhead.

Moreover, the paper extends the paper to non-Gaussian states through the introduction of $t$-doped Gaussian states. These are states prepared by applying Gaussian unitaries and a controlled number of non-Gaussian local operations. The authors show that for a small number of non-Gaussian operations, efficient tomography is still achievable. This is significant because it bridges the gap between purely Gaussian states and more general non-Gaussian quantum states typically encountered in practical quantum information processing tasks.

The implications of this paper are substantial. Efficient tomography of Gaussian and slightly non-Gaussian states underlines the feasibility of such measurements in practical quantum devices. It suggests that the inherent complexities of CV quantum states can be managed effectively within constrained resources, advancing developments in the field of quantum computation and quantum communication. Future research may build on these foundations, potentially exploring experimental implementations and further reducing the computational overhead associated with CV state learning.

This work presents substantial theoretical advancements in understanding quantum state tomography's sample complexity for continuous-variable systems, contributing significantly to the field of quantum information science. The methods outlined provide a pathway for practical applications in a wide array of quantum technologies relying on continuous-variable frameworks.